Abstract

We formulate the characteristic problem for one fixed retained Einstein–scalar kernel, keeping all seven internal bookkeeping rows, all three mixed-vertex terms, and the grouped crossed-line difference. The actual eight-relative-variable smoothing is represented exactly by a compact, test-dependent translation distribution.

The main result is a conditional decision theorem. A specified finite logarithmically homogeneous origin germ of the complete translation distribution, with a uniformly Hölder continuous remainder, determines every nonsummable coefficient and the finite part. It yields quantitative bounds for both the family remainder and its logarithmic scale derivative. The raw family converges exactly when its complete nonsummable coefficients vanish; otherwise one fixed compact joint test detects unboundedness. Coefficient uniqueness makes the conclusion independent of internal presentations. Support, profile independence, admissible subtraction, and normalization are treated separately.

We reproduce the finite source rule and density derivative, derive the shifted characteristic equations with their correct second-slot sign, and retain the full coordinate density. An appendix evaluates an exact nonphysical convolution diagnostic, including all inverse powers and an order-eight remainder.

The supplied materials do not instantiate the designated physical crossed covariances or verify the complete origin-germ hypothesis for them. Thus the conditional theorem is proved here, but no evaluated convergence, failure, or admissible-extension decision for the designated physical graph is asserted.